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\title{\vskip -40pt           % move title up
\huge The Paraboloid 
\footnote{This file is from the 3D-XploreMath project. You can find it on the web by searching the name.}
\vskip -20pt  %reduce space between ttle and authorname.
}

\author{H.K.}

\begin{document}

\maketitle

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\LARGE


\cl{ See in Documentation: About Quadratic Surfaces.}
\centerline{\includegraphics{Paraboloid.png}}

\LF
The Paraboloid in 3D-XplorMath is parametrized as
$$
  x = aa\cdot u\cdot \cos(v), \hskip3mm y= bb\cdot  u\cdot \sin(v), 
  \hskip3mm z = cc\cdot  u^2,
$$
with the default $aa= bb=1,\ cc=0.4$.
\Lf

The paraboloid is shown together with a few rays parallel  to the z-axis, the axis of
revolution symmetry of this surface. These rays are reflected in the surface and
continued until they meet in the focal point of this paraboloid. This image looks
somewhat like the reflector of a car headlight together with the rays from the
light bulb, reflected into parallel rays. The {\tt default Morph} varies $cc$ so that
the image changes from a headlight reflector to a satellite antenna, with incoming
parallel rays concentrated on the receiver at the focal point of the antenna.
\Lf
The entry {\tt Remove Focal Rays} in the Action Menu returns to the standard
rendering for surfaces. Only in {\tt Wireframe Display}  can one switch on the focal
rays in the Action Menu.
\Lf
For geometric arguments concerning the focal point see: {\tt Parabola} in the
Plane Curve category.


\bye